Question: New visitors to a certain website have a $0.15$ probability of creating an account on their first visit. Let $V$ be the number of new visitors who visit the site until the first one of them creates an account. Assume that each visitor's decision to create an account or not is independent. Find the probability that the $6^{\text{th}}$ new visitor is the first to create an account. You may round your answer to the nearest hundredth. $P(V=6)=$
Answer: Without a fancy calculator For each new visitor: $P({\text{account}})=0.15$ $P(\text{not}})=0.85$ If the $6^{\text{th}}$ new visitor is the first to create an account, there must be $5$ new visitors who do not create an account followed by the $1$ who does. $\begin{aligned} P(V=6)&=P(\text{NNNNN}}{\text{A}}) \\\\ &=(0.85})(0.85})(0.85})(0.85})(0.85})({0.15}) \\\\ &=(0.85)^5(0.15) \\\\ &\approx0.066556 \end{aligned}$ $P(V=6) \approx 0.066556 \approx0.07$